Delivery is everything. If poison or other damage has these qualities then I allow double damage on a critical: 1) The damage must apply immediately upon scoring the hit. 2) The damage must be expressed as a number of dice. That's it.
Agreed. And I think both of your points can be clearly pulled from the RAW in the PHB.
If a save is allowed to halve the damage, the test is still met. OTOH if the extra damage only applies IF a save is failed then the test is NOT met. It keeps things simple, clear cut and consistent, and does not discriminate based on the source of the damage (ability, equipment, magic, etc.)
I don't follow you on this one. Can you give an example? Most savings throws deal half damage on a successful save. Are you saying that Lightning Arrow, because it triggers a save (that does half damage if successful) would auto-crit on a hit, if the hit that caused the Lightning Arrow to burst was a crit?
I would see that Lightning Arrow as a separate effect triggered off the hit (or miss) which triggers a separate d20 roll which would not be subject to critical hits because a) it is a separate roll and b) it is not an attack roll. This actually meets your 1st rule: the damage does not apply immediately on the critical hit (because you still need to roll to see if the effect was saved or not).
With regard to witch bolt I would not double subsequent round damage because no hit roll was required. Without a "to hit" roll there can be no critical hit.
What makes Witch Bolt an interesting edge case is there is only one d20 roll (a range spell attack roll). If you miss, there is no subsequent damage on following rounds so the next round damage is directly caused by the initial hit. There's also concentration on the spell so it is conceivable that the character is "keeping his initial hit going." In the end we ruled as a group that it doesn't make sense because it is a separate damage roll. With Sneak Attack you might roll all your damage at once, but with With Bolt you wouldn't. On a hit, subsequent rounds simply "auto-hit" with no roll, similar to Magic Missile, so the crit doesn't carry over.
On that note I would play a Rogue with Sneak Attack and a Poisoned Dagger that reads "If you hit with this dagger the target makes a DC 17 CON savings throw or takes an additional 2d6 poison damage.":
1) Roll a d20 to see if he hits
2) It's a critical hit! Roll twice as many dice for the dagger and sneak attack damage (2d4 + 4d6 + modifiers)
3) Roll a d20 to see if the target saves
4) He failed the save, roll 2d6 for the poison damage.
Since it is a separate damage roll that happens on a separate d20 roll, I would not double the damage dice. I think this example is the easiest way to illustrate how I think the rules in the PHB is intended. In particular the d20 roll in step 3 was not an attack roll so it isn't subject to critical hits.
If the Poisoned Dagger read slightly differently: "If you hit with this dagger the target makes a DC 17 CON savings throw or takes an additional 2d6 poison damage or half on a successful save" I would still run the rolls the same way, it's just now the effect does half damage on a successful save.